Keith Devlin teaches the theorem that there are infinitely many primes in his course on Mathematical Thinking.
Devlin’s proof is easy to follow. He has a good style. Viewers may wish to compare his method to that of the Khan Academy video style with Wacom.
The above are Khan Academy videos relating to simple number theory.
What grade level could we teach the theorem including its proof that there are infinitely many primes?
According to Marco Flores at Zunal.com this is 6th to 8th grade level.
Page 12 of following discusses grade 6 to 8 standards in math.
A reference on teaching the infinitely many primes is here.
WORKING GROUP 4. Argumentation and proof 536
Argumentation and proof 537
Maria Alessandra Mariotti, Kirsti Hemmi, Viviane Durrand Guerrier
Indirect proof: An interpreting model 541
Samuele Antonini, Maria Alessandra Mariotti
Mathematics learning and the development of general deductive reasoning 551
Michal Ayalon, Ruhama Even
How to decide? Students’ ways of determining the validity of mathematical statements 561
Orly Buchbinder, Orit Zaslavsky
Cabri’s role in the task of proving within the activity of building part of an axiomatic
Leonor Camargo, Carmen Samper, Patricia Perry
Some remarks on the theorem about the infinity of prime numbers 581
Ercole Castagnola, Roberto Tortora
Proofs problems in elementary number theory: Analysis of trainee teachers’ productions 591
Annalisa Cusi, Nicolina A. Malara
Relationship between beginner teachers in mathematics and the mathematical concept of
Using the Van Hiele theory to analyse the teaching of geometrical proof at grade 8 in
Liping Ding, Keith Jones
Analysis of conjectures and proofs produced when learning trigonometry 622
Jorge Fiallo, Angel Gutiérrez
Analysis of the teacher’s arguments used in the didactical management of a problem
solving situation 633
Structural relationships between argumentation and proof in solving open problems in
Mathematical proof: Teachers’ beliefs and practices 653
The mental models theory of deductive reasoning: Implications for proof instruction 665
Andreas J. Stylianides, Gabriel J. Stylianides
Reviewing textbook proofs in class: A struggle between proof structure, components and
=Page 47 Infinitely Many Primes proof
“We want to begin comparing three versions of the proof: the original Euclid’s one
(Proposition 20, book IX of Elements), as reported in (Heath, 1956); the “modern”
(1925) version of the same author; and that used today in mathematics texts.” page 49.
=Page 49 excerpt
THE THEOREM ON THE INFINITY OF PRIME NUMBERS
The theorem on the infinity of prime numbers is one of the most famous and of
the most “beautiful” in the history of mathematics. Several proofs have been
produced (see for instance the website (3)), but the best known, modelled on Euclid’s
original proof, is surely the most easily understood, a striking example of simplicity
and elegance. In spite of that, this proof appears much more obscure for students than
we could think at first sight. A deep and careful analysis of the proof and of its
didactical implications is presented in (Polya, 1973). After that, many authors have
focused their attention on the difficulties involved in the contradiction argument
employed in the proof, e.g. (Reid & Dobbin, 1998). Other authors have underlined
the logical subtleties, all but easy to be understood, involved in such kind of
reasoning (Antonini, 2003), (Antonini & Mariotti, 2006); or the necessity to enter an
“imaginary” world, where the usual rules of logic can be put in doubt (Leron, 1985).
(For more references, see the quoted papers). In particular, Leron notes how the
“distance” between the assumption a contrario and the conclusion causes the total
loss of all the constructions performed in the intermediate steps, erroneously
perceived as meaningless.
They also consider this problem:
==Page 59 of pdf
PRESENTING THE PROPOSED PROBLEM
The problem at stake is the following: “Suppose that a is a non null natural number.
If a is divisible neither by 2 nor by 3, then a*a-1 is divisible by 24”. This problem is
taken from the textbook, aimed at 15-16 years old students, “Matematica come
scoperta” (“Mathematics as discovery”) by G. Prodi (1979). This textbook was
thought and written in a research-based environment and it is still very innovative.
The problem was given to 54 trainees with different university backgrounds (27
mathematics graduates, 3 physics graduates and the remaining 24 biology, geology,
chemistry and natural sciences graduates) in the initial phase of the Mathematics
Education training courses. Trainees were supposed to solve, in 45 minutes, the
assigned problem, describing the different proving strategies they tried to follow in
the solution process and pointing out both obstacles and difficulties they met.
Let’s Play Math gets into it.
It links here:
They try to fit the proof to the Twitter span of 140 characters.
Kentucky Grade 6 standards
You can add Euclid to this search.
Proof by Contradiction video
Infinitely many primes 8 minutes in.
He goes through the steps more slowly than Devlin.
The above does theorem (n+1)^-n^2 = n+(n+1)
= 2n+1 = 2n’-1
Notice how easier it is if we use prime notation.
n’ n’ = (n+1)n’ = nn’ + n’ = n (n+1) + n’ = nn + n + n’
n’ n’ – nn = n + n’